This document discusses several factors related to gas/steam flow in turbomachines: 1. Cavitation and sonic/supersonic flows are not concerns if the medium is already a gas/steam. 2. Flow velocity should be kept below the sonic velocity to avoid losses from shock waves. The sonic velocity depends on temperature and is lowest at the suction end. 3. Parameters like the inlet number ε, discharge number ε2, and suction diameter can be used to relate flow properties and avoid cavitation or sonic velocities. Their values depend on factors like the shape number Nshape.

1. Gas/Steam Medium
1

2. • If the medium is already a gas/steam, the phenomenon that
the flow medium will change from the liquid to the gas phase
does to occur, thus no cavitation has to be considered.
• For lower temperature, the sonic velocity is low. In order to
avoid supersonic and sonic flows the flow velocity has to
lower. But the flow velocity may be desired to be higher.
• Above sonic velocity(supersonic flow) there is an occurrence of
shock waves which represent losses.
2

3. • The nearness of the flow velocity to the sonic velocity a may
be expressed using the Mach Number.
• IN general, flow conditions where M=1 and normally also
those with M>1 (supersonic flow) have to be avoided.
• The sonic velocity of a gas is:
• Since for a given gas k and R are constant, the locally
prevailing temperature determines the sonic velocity a, the
danger or reaching sonic velocity is greatest where the
temperature is the lowest.
• In general, this is at the suction end of the turbomachine.
a
C
M 
kRT
P
k
a 


3

4. Compressors
• The sonic velocity is lowest at the suction end of the impeller,
in the case of multi-stage compressors, at the suction end of
the first impeller as here the temperature is lowest.
• The criteria to limit the locally highest velocity below the sonic
velocity may be expressed as follows:
• Similarly to avoiding cavitation, an optimum angle βoa can be
determined:
8
.
0
7
.
0 to
a
W
M oa

 The lower value refers to impeller with thick vanes
The higher value to those with thin vanes
    2
2
max 1 oa
W
W 


Where Wmax = locally highest velocity in the vane
channel near suction end
Experimental value: λ ≈ 0.2 to 0.3
4

5.  Assuming no pre-rotation (δr=1) and the value of λ the
optimum angle βoa is:
 The optimum angle which avoids sonic velocity best has value
of nearly twice of optimum angle avoiding cavitation best
 By applying the above criterion the velocity in the channel
near the suction edge is kept lowest. As the velocity decreases
the frictional loss also decreases which results a higher
hydraulic efficiency.
  '
30
32
'
10
32 0
0
to
opt
oa 

5

6. The Suction Diameter-The Inlet Number ε
and the Discharge Number ε2
6

7. • The suction diameter has to be chosen so that βoa obtains its
desired value with regard to avoiding cavitation or sonic
velocity or with regard to obtaining lowest friction loss at the
vane suction edge.
• The following optimum values of βoa were mentioned in the
previous chapters:
1
:
:
;
35
:
20
:
17
0
0
0


















r
opt
oa
opt
oa
opt
oa
if
vanes
rotor
of
edge
suction
at
loss
friction
lowest
obtaining
ne
turbomachi
all
velocity
sonic
avoiding
Compressor
woa
smallest
cavitation
avoiding
turbine
cavitation
avoiding
pump




7

8. • The suction diameter D1a and the angle βoa are interrelated to
each other by the velocity Coma as follows
• A relation between Com and D1a can be found from the
velocity triangle at point o:
 
a
o
o
om D
f
A
where
A
V
C 1
'


Cross sectional area at point o
Perpendicular to Com
2
1
1











s
n
r
D
d
k

a
r
om
oua
a
om
oa
U
C
C
U
C
1
1
tan

 


8

9. It follows
and
ality
proportion
of
factors
f
,
f
where
tan
2
tan
2
1
2
1
1
1
1 S
a
s
oa
a
r
oa
a
r
oma D
f
D
with
C
f
D
U
C 


 




2
1
1
1
4
'
'
S
S
S
oma
D
k
V
f
A
V
f
C
f
C




3
2
1
2
3
2
1
tan
'
4
tan
'
8
r
oa
oa
r
S
f
f
n
k
V
f
k
V
f
D








9
oa
S
r
oa
a
r
oma
D
f
U
C 



 tan
2
tan 2
1 

oa
S
r
S
D
f
D
k
V
f




tan
2
4
'
2
2
1


10. • The suction diameter Ds may also be determined using the
following dimensionless numbers:
• Knowing the value of ε, ε2 , the suction diameter Ds follows
from
• The values of ε and ε2 are functions of the shape Number
Nshape as can be noted from the following derivation:
Pumps and
compressors
Inlet Number
Y
om
om
C
C
Y
C


2

turbines Discharge Number
2
2
2
2
2 Y
om
om
C
C
Y
C



2
1
1
1
S
S
o
om
D
A
A
C 


10

11. V
f
V
if
f
n
k
V
f
f
n
f
n
D
f
D
f
C
oa
r
oa
r
oa
S
r
oa
S
r
oma
3
3
2
2
3
1
2
2
2
'
tan
4
tan
tan
tan
2
















and oma
om
oa
r
om
C
C
if
Y
k
V
n
f
f
f
Y
C


 3
2
/
3
2
2
3
2
2
1
2
tan
4
2
1
2




where 2
2
/
3
2
shape
N
Y
V
n

3
/
2
3
3
2
2
1
tan
2
4






 shape
oa
r
N
k
f
f
f




3
/
2
3
3
2
2
1 tan
64
.
1 





 shape
oa
r
N
k
f
f
f 


11

12. • The Inlet number for pumps and compressors, assuming δr=1 and
k = 1-(dn/Ds)2≈0.8 and the common range βoa =14 to 380 is
 
3
/
4
3
/
2
2
2
1
2
tan
70
.
2 





 shape
oa
r
N
k
f
f
f 


   
inlet
rotor
at
loss
friction
low
velocity
Sonic
Cavitation
f
f
f
where
N
f
f
f
to shape
/
1
.
1
50
.
1
70
.
0 3
3
2
2
1
3
/
2
3
3
2
2
1





As far as the slow-running rotor with Nshape < 0.1 and k =1 is
concerned the value of ε may be taken independent of Nshape as
inlet
rotor
at
loss
friction
low
velocity
Sonic
Cavitation
to
/
5
.
0
1
.
0




12

13. • The values δr and k of water turbines of normal designs are
δr≈1 Francis, Kaplan Turbine
K ≈ 0.8, Kaplan Turbine, k ≈1 Francis Turbine
• The following table shows some values of ε2 taken from actual
designs.
Francis Turbines Kaplan T Dim.
Nshape 0.063 0.065 0.123 0.210 0.34 0.52 0.70 1
ns 70 94 141 237 387 592 797 (metric)
nq 21 28 41 70 114 174 234
(metric)
ε2 0.032 0.032 0.048 0.096 0.152 0.331 0.486 1
βoa 30 23.5 21.3 21.3 18.6 19.4 19.2 degree
13

14. • Faster running turbines have higher values of ε2. In case of the
Kaplan Turbines ε2 may account up to 0.5.
• This means that the kinetic energy Co
2 /2 with which the water is
discharged from the rotor is equal to 50 % of the available energy Y.
• All this kinetic energy would be lost if no draft tube were provided. A
draft tube, however, will ‘recover’ part of this kinetic energy.
• As ε2 of fast running turbines is large, these turbines have to be
designed with very effective draft tubes. For this reason Kaplan
turbines are mostly designed with elbow type draft tubes.
14

15. Number of Vanes
15

16. • If the number of vanes is small, each vane is loaded much and,
hence the pressure difference between both sides of the vane is
high.
• This leads to non uniform velocity distribution in the vane channel
and very high local velocities may occur. Thus also cavitation or
sonic velocity is more likely to occur.
• If the number of vanes is high the vane channel becomes narrow
and consequently the friction loss is high.
• Thick vanes do not allow to install many vanes. Generally the
thickness of the vanes should be as small as possible observing,
however the strength of the vane material, the vibration of the vane
and proper profiling if desired.
16

17. • Number of vanes
• Pumps and water turbines have mostly cast vanes, radial-flow blowers have
sheet metal vanes.
• Axial- Flow
• The above formulas are also applicable to determine the number of
vanes of guide vanes.
2
sin
2 2
1 
 

e
r
k
Z m
where e = length of the mean stream line in
meridian section measured between
vane in-and outlet
rm = radius of the middle of the line e
k = empirical factor
= 5 to 6.5 cast vanes
= 6.5 to 8(to 12) sheet metal vanes
 
2
sin
,
2
1 2
1
1
2
2
1
2
1
1
2

 







r
r
r
r
k
Z
thus
r
r
r
and
r
r
e m
17